I have solved this puzzle. I'm not sure if I used an "advanced technique," or not.

I'll just describe the position I reached in general terms for now. After about 12 moves (I'm not sure exactly how many -- I'll have to go back and retrace the early steps) I reached a position that had three noteable characteristics:

1. the only possibilities at r4c4 were {5, 7};
2. there were several columns and rows where "9" could only appear in two places; and
3. the set {2, 6, 8} already appeared in column 4.

What I noticed at this point was that placing a "5" in r4c4 made it impossible to complete the top middle 3x3 box, from which I inferred that r4c4 = 7 -- the rest of the puzzle was relatively straightforward. dcb

At this point the grid looked like this -- I've only entered the possibilities that I had pencilled in, leaving the more complex possibilities blank. I have also marked all the spots where a "9" can possibly appear with an asterisk ("*").

At this point I noticed that if I placed a "5" at r4c4 the middle psrt of column 5 (that is, r4c5, r5c5, and r6c5, in the middle box) would be reduced to the triplet {1, 4, 6}, substantially simplifying the analysis of columns 4, 5, and 6. So I traced out the following forced moves:

But now there are only four cells (r1c4, r1c6, r2c4, and r2c6) left open in the top middle 3x3 box, and these four cells must contain the values {1, 2, 6, 8} -- since the values {2, 6, 8} already appear in column 4, this clearly cannot be done. Therefore the hypothetical move r4c4 = 5 can be eliminated, and we must have r4c4 = 7. The rest of the puzzle is easily solved from there.

There's always more than one way to skin a cat! dcb

Last edited by David Bryant on Fri Sep 30, 2005 6:28 pm; edited 1 time in total

This IS a tricky sudoku. You can do it by finding pairs, etc. AND by the X-wing that Someone_Somewhere gives: at one stage, 8 can go in rows 2 and 8 in the columns 6 and 7 only. So logically you can have an 8 nowhere else in these columns, which turns out to be enough to solve it all without the need for XY-wing, by later finding sneakily hidden multiples.

I decided to give your puzzle a try, and went as far as I could. In looking at your steps to solve, I don't understand what a "naked pair" is. You identified 1,8 as a naked pair in r8c6 and r8c7. Please explain this to me. Thanks!

I decided to give your puzzle a try, and went as far as I could. In looking at your steps to solve, I don't understand what a "naked pair" is. You identified 1,8 as a naked pair in r8c6 and r8c7. Please explain this to me. Thanks!

A "naked pair" is a set of two numbers that must appear in two cells -- either in a row, or in a column, or in some 3x3 box. Here's an example.

Code:

1 2 3/7 4 5 6 3/7 8 9

In this hypothetical example, we already know how to place 7 digits in a row, but the position of the other two numbers {3, 7} is still indeterminate.

This can be a useful piece of information, as in the following example.

Code:

1 2 3/7 4 ? 6 3/7 8 9

The cell marked "?" is not yet known -- we can immediately see that it must be a "5" because of the "naked pair" {3, 7}.

This kind of set-up is called a "naked pair" to distinguish it from a "hidden pair" -- here's an illustration of a "hidden pair":

Code:

1 2/4/5/6 3 2/4/6/9 5/6/9 5/6/9 7 8 5/6/9

In this example the "hidden pair" appears in the second and fourth positions. We can see it there because the two digits {2, 4} cannot appear in any other spots in the row, except the second and fourth positions. But it's also "hidden" because there are apparently some other values that _might_ appear in the 2nd and 4th positions.

Another way to spot the "hidden pair" is to notice the "naked triplet" {5, 6, 9} appearing in the 5th, 6th, and 9th positions. Since those three cells must contain the 3 digits {5, 6, 9}, we can eliminate {5, 6, 9} from the 2nd and 4th positions, exposing the "hidden pair."

Thanks, David for explaining that to me! I don't see the naked pair in row 8. I have a 1,8 pair in row 8 in r8c6, r8c7 and r8c8. I've been following SomeoneSomewhere's solution and he must have put a 6 in r8c8 in order to get the naked pair. I don't see how the 6 gets there. In box 9 I have a 6 in r8c8 and r9c8.

I found the naked pair! I had put a 3 in r8c6 which was a mistake since column 6 already had a 3 in it! It's amazing that one mistake can throw off the whole puzzle! I'll see if I can finish this bad boy, otherwise you may hear from me again! I appreciate your patience and excellent teaching!

Do you have an example that can be solved with "quadruples" but not with "pairs" or "triples"? Could be in paralel with X-wing, XY-wing, coloring or some more advanced than "quadruples".

Thank you in advance,

see u,

I'm not sure if I have such an example, or not. I did notice that the subject of "naked quadruplets" came up twice recently in posts from Chobans -- one of those might be what you're looking for. Here are the links.

Do you have an example that can be solved with "quadruples" but not with "pairs" or "triples"? Could be in paralel with X-wing, XY-wing, coloring or some more advanced than "quadruples".

Thank you in advance,

see u,

I did this puzzle yesterday from USA Today. I needed to use the X Wing technique to solve it. I'm not a whiz kid like you and David are so perhaps this one can be solved without advanced techniques!
[/url]http://puzzles.usatoday.com/sudoku/

2 not in r1c8, it is in r3c8 or r3c9 (Row on 3x3 Block interaction)
2 not in r1c9, it is in r3c8 or r3c9 (Row on 3x3 Block interaction)
2 not in r2c8, it is in r3c8 or r3c9 (Row on 3x3 Block interaction)
2 not in r2c9, it is in r3c8 or r3c9 (Row on 3x3 Block interaction)
5 not in r8c2, it is in r7c2 or r7c3 (Row on 3x3 Block interaction)
7 not in r8c2, it is in r7c2 or r7c3 (Row on 3x3 Block interaction)
7 not in r9c2, it is in r7c2 or r7c3 (Row on 3x3 Block interaction)
7 not in r9c3, it is in r7c2 or r7c3 (Row on 3x3 Block interaction)
3 not in r6c9, Hidden Pair 2 5 in r3c9 and r6c9 (in Column)
4 not in r6c9, Hidden Pair 2 5 in r3c9 and r6c9 (in Column)
6 not in r6c9, Hidden Pair 2 5 in r3c9 and r6c9 (in Column)
9 not in r6c9, Hidden Pair 2 5 in r3c9 and r6c9 (in Column)
9 in r5c9 - Unique Vertical
... (and the rest is no problem)

3 in r3c5 - Sole Candidate
6 in r2c5 - Sole Candidate
2 in r2c6 - Unique Horizontal
7 not in r2c8, it is in r1c8 or r1c9 (Row on 3x3 Block interaction)
7 not in r2c9, it is in r1c8 or r1c9 (Row on 3x3 Block interaction)
3 not in r8c1, it is in r9c1 or r9c2 (Row on 3x3 Block interaction)
3 not in r8c2, it is in r9c1 or r9c2 (Row on 3x3 Block interaction)
5 not in r7c1, it is in r9c1 or r9c3 (Row on 3x3 Block interaction)
5 not in r8c1, it is in r9c1 or r9c3 (Row on 3x3 Block interaction)
5 not in r8c3, it is in r9c1 or r9c3 (Row on 3x3 Block interaction)
3 in r9c2 7 in r1c8 - Unique Vertical
1 not in r8c3, it is in r7c2 or r8c2 (Column on 3x3 Block interaction)
6 not in r8c3, it is in r7c2 or r8c2 (Column on 3x3 Block interaction)
6 not in r9c3, it is in r7c2 or r8c2 (Column on 3x3 Block interaction)
9 not in r2c1, it is in r2c2 or r3c2 (Column on 3x3 Block interaction)
5 not in r7c4, it is in r7c5 or r8c5 (Column on 3x3 Block interaction)
5 not in r8c4, it is in r7c5 or r8c5 (Column on 3x3 Block interaction)
7 not in r8c6, it is in r7c5 or r8c5 (Column on 3x3 Block interaction)
2 not in r8c9, it is in r7c8 or r8c8 (Column on 3x3 Block interaction)
4 not in r2c7, it is in r2c8 or r3c8 (Column on 3x3 Block interaction)
8 not in r1c7, it is in r2c8 or r3c8 (Column on 3x3 Block interaction)
8 not in r2c7, it is in r2c8 or r3c8 (Column on 3x3 Block interaction)
8 in r1c4 6 in r9c6 - Unique Horizontal
8 in r8c6 - Unique Vertical
1 in r3c6 - Unique Vertical
9 in r2c4 - Unique in 3x3 block
9 in r3c2 - Unique in 3x3 block
8 in r2c2 5 in r3c9 - Sole Candidate
4 in r2c8 4 in r3c3 - Sole Candidate
8 in r3c8 - Sole Candidate
4 not in r4c4, it is in r4c6 or r6c6 (Column on 3x3 Block interaction)
4 not in r6c4, it is in r4c6 or r6c6 (Column on 3x3 Block interaction)
5 not in r7c7, it is in r7c8 or r8c8 (Column on 3x3 Block interaction)
5 not in r8c7, it is in r7c8 or r8c8 (Column on 3x3 Block interaction)
6 in r7c7 - Sole Candidate
9 in r1c7 1 in r7c2 - Sole Candidate
6 in r1c9 4 in r7c4 6 in r8c2 3 in r8c7 - Sole Candidate
1 in r2c7 1 in r8c4 9 in r8c9 - Sole Candidate
3 in r2c9 - Sole Candidate
4 in r8c1 - Unique Horizontal

After 27 moves I arrived at a position where the only method I could apply was to make an outright guess. I tried using a couple of different computer programs on it then, and those also had to resort to guessing.

Can anyone spot a logical way through this puzzle? Does USA Today typically publish puzzles that require trial and error? dcb

You're right! I guess all the zeroes threw me off ... I didn't notice the three columns filled up in the three middle 3x3 boxes with all those zeroes in there!

I also tried to apply "Nishio", but started from the {1, 8} possibility in r5c3, just because it looked odd. Why did you start from r4c1? dcb

PS So you and I were working on the same puzzle, but are we sure it's the one Louise56 was talking about? Is this the one you were talking about, Louise?

Hi David,

The one I did from USA today was the Sept 30 one, which is the same one you and Someone worked on. I can't remember all my steps, I'll have to go back. I did an X wing three times as I remember. The X Wing numbers were all four fairly close to each other, but that helped me elimate others. I'll go through it again and write a solution. Can you tell me what a "Nishio" is?